In most worlds, the goblins and elves will eventually outbreed everyone else because they are immortal but also reproduce. Elves are less likely to outbreed goblins by dint of their wooden weapons and armor, but they can still overwhelm if goblins aren't around, given enough time and space.
So here is a radical suggestion: Immortal races don't have children! The cultures of the elves can be continued with animal-people immigrants. Elves are already freinds to animals, so it follows that animal-people would be the logical mortal choice for elvish culture. The animal people would be their surrogate children, and those born in elvish lands would take on the culture and mores of the elvish culture. Elves should be given preferential treatment for positions of authority in this culture, as the immortal "Elder Race" overseeing generations of animal-people as they keep the glades and forest homes of the Elvish nations in good order.
Goblins by their very nature are violent and doomed to fight one another as well as serve their demon overlords. Death comes to them quickly (but apparently not as quickly as they breed). However, they also kidnap other races and this is where they should be able to gain their "new recruits". In stead of making new goblin children, it's the kidnapped "immigrants" and their children who become the replacements for their goblin masters. Have the immortal goblins fight with each-other, but get preferential treatment when it comes to positions of authority. Kidnapped races and their children only get positions of power if there is no other goblin able to take it.
This will eventually lead to the weaker or less treacherous Elves and Goblins being killed off in one way or another and most of the remaining ones either being high priests or nobility of some sort, perhaps pursuing esoteric concepts like science or magic or writing endless books. In later years, elvish or goblin weapon-masters become late-game challenges, each with thousands of kills under their names.
The mortal races of kobolds, animal people, humans and dwarves would reproduce and die like normal, eventually taking on different cultures as they immigrated to and were transformed into goblin or elvish cultures. Wars or alliances and the ability to affect cultural change in that way would also become much more feasable.